## Introduction

Collision between electrons of a metal leads to a T-square resistivity. Postulated in 1930s by Landau and Pomeranchuk1 and independently by Baber2, this feature has been widely documented in elemental3 and strongly correlated4 metals. At sufficiently low temperature, their resistivity (ρ) follows this simple expression:

$$\rho ={\rho }_{0}+A{T}^{2}$$
(1)

The residual resistivity, ρ0, depends on defects but, A is an intrinsic property of the metal, found to scale with the electronic-specific heat3,4 in dense metals (i.e., those having roughly one carrier per formula unit). The quadratic temperature dependence of the phase space is a consequence of the fact that both participating electrons reside within a thermal window of the Fermi level.

In absence of a lattice, an electron–electron collision conserves momentum and cannot degrade the charge current. To generate finite resistivity, such collisions should transfer momentum to the lattice. There are two known mechanisms: either it is because there are multiple electron reservoirs unequally coupled to the lattice2,3, or because the collision is an Umklapp event5,6. In the first case, the two colliding electrons have distinct electron masses2. Momentum transfer between these two distinct reservoirs sets the temperature dependence of resistivity, and the mass mismatch causes momentum leak to the lattice thermal bath. In the second case, one of the two colliding electrons is scattered to the second Brillouin zone and returns to the first one by transferring a unit vector of the reciprocal lattice (G) of momentum to the lattice5,6.

The observation of a T2 resistivity in dilute metallic SrTiO3 indicated, however, that our understanding of the microscopic foundations of this ubiquitous phenomenon is unsatisfactory7. SrTiO3 is a cubic perovskite at room temperature. It is a quantum paraelectric and becomes a dilute metal upon introduction of a tiny concentration of mobile electrons8. Three concentric conducting bands centered at Γ point of the Brillouin zone are successively filled9. The quadratic temperature dependence of its electrical resistivity10,11 persists7 even when its Fermi surface shrinks to a single pocket12 and none of the two mechanisms operate. However, it was more recently suggested that this enigmatic T-square resistivity may be caused by exotic mechanisms such as scattering by magnetic impurities13 or by two soft transverse optical phonons (See Supplementary Discussion)14,15. Such soft phonons are known to play a decisive role in transport properties of the system, at least at high temperatures16,17,18,19.

Here, we report on another dilute metal, doped Bi2O2Se. We show that it displays T2 resistivity whilst the Hall carrier density (n) changes by two orders of magnitude and in absence of interband and Umklapp scattering. Moreover, there are no soft phonons and the T-square resistivity is restricted to temperatures well below the degeneracy temperature. The e–e origin of this T-square resistivity is unambiguous. Comparing the evolution of A with n and EF in Bi2O2Se and SrTiO3, we uncover a universal scaling between A and EF in dense and dilute Fermi liquids. Our results imply that a proper microscopic theory of the link between electron–electron scattering and T-square resistivity is still missing.

## Results

### First-principle calculations of Bi2O2Se single crystal

Stoichiometric Bi2O2Se is a layered semiconductor with tetragonal crystal structure (anti-ThCr2Si2 phase) at room temperature20, shown in Fig. 1a. Available Bi2O2Se single-crystals are metallic with extremely mobile carriers21,22,23. The insulator is doped by unavoidable defects, such as Se or O vacancies and Se–Bi antisite defects24,25. According to density functional theory (DFT) calculations24, the conduction band is centered at the Γ point of the Brillouin zone, following a parabolic dispersion (Fig. 1b and Supplementary Fig. 1). This has been revealed by angle-resolved photoemission spectroscopy (ARPES) measurements21,26. The Fermi surface is an elongated ellipsoid (Fig. 1a) seen by quantum oscillations26. The comparison of crystal and band structure between Bi2O2Se and SrTiO3 is summarized in Supplementary Table 1.

Our DFT calculations of phonon spectrum for the tetragonal phase of Bi2O2Se are in agreement with what was reported previously27,28. Figure 1c presents the phonon dispersion at ambient pressure. The absence of imaginary frequencies implies that the tetragonal phase is dynamically stable. Figure 1d–f shows the evolution of the calculated local potential and phonon frequencies with hydrostatic pressure. Clearly, the single-well local potential at ambient pressure is different from a quantum paraelectric, where the local potential is a shallow double-well and the ferroelectric phase is aborted due to quantum tunneling between two local minima of the well. In a quantum paraelectric, the soft mode is very sensitive to hydrostatic pressure. For example, in PbTe a well-known quantum paraelectric, the soft mode frequency almost triples by reducing the lattice constant (a) by 1%29. In contrast, all three optical modes change moderately by reducing a, in Fig. 1f. The lowest mode (TO1) hardens only by 20% by a similar reduction of a. We also note that a large negative pressure (≈−5 GPa) is required to make the system ferroelectric. These observations rule out the proximity to a ferroelectric instability, excluding the presence of soft phonons.

### Quadratic temperature dependence of resistivity

Figure 2a shows the temperature dependence of resistivity ρ(T) in our Bi2O2Se samples with various n (See Supplementary Fig. 2, Supplementary Fig. 3 and Supplementary Table 2 for more information). We note that our data (obtained for n < 1 × 1019 cm−3) smoothly joins what was recently reported for n = 1.1 × 1019 cm−322. Upon cooling from room temperature to 1.8 K, resistivity decreases by two orders of magnitude, comparable to what is seen in doped quantum paraelectrics such as SrTiO3, KTaO3, and PbTe16.

Panels b–e of Fig. 2 document the low-temperature quadratic temperature dependence of resistivity. It persists down to the lowest temperatures in all the samples. Equation (1) holds below a characteristic temperature (dubbed Tquad). As seen in Fig. 2f, Tquad is an order of magnitude lower than TF, the degeneracy temperature of the fermionic system (extracted from our study of quantum oscillations described below). The relevance of TquadTF inequality in Bi2O2Se contrasts with what was seen in SrTiO37. Only well below the degeneracy temperature, one expects the phase space of the fermion-fermion scattering to be quadratic temperature. Therefore, one reason to question the attribution of T-square resistivity to e–e scattering, which was raised in the case of SrTiO3, is absent.

Figure 2g presents the evolution of the T2 prefactor (A) with n for Bi2O2Se. For comparison, the relevant data for doped SrTiO3 and KTaO3 are also shown. In both cases, A decreases with increasing n. While A(n) is more fluctuating in Bi2O2Se than in SrTiO3, the two slopes ($$\frac{n}{A}\frac{{\rm{d}}A}{{\rm{d}}n}$$) are close to each other. At similar n, the magnitude of A differs by more than one order of magnitude lower. We will see below that this arises because of the difference in the magnitude of the effective electron mass, which sets the Fermi energy at a given carrier density.

Thus, Bi2O2Se is the second metallic system in which T-square resistivity is observed in presence of a single-component and small Fermi surface. The absence of multiple pockets leaves no place for interband scattering. The smallness of the Fermi surface excludes Umkalpp events (See below for quantitative discussion).

Two features make the theoretical challenge thrown down by Bi2O2Se more solid than the one presented by SrTiO3. First of all, since TquadTF, one objection to associating T-square resistivity and e–e scattering vanishes. Moreover, the absence of any exotic soft phonons, rules out their possible role as scattering centers14,15.

Having shown that T-square resistivity is present in Bi2O2Se like in SrTiO3 without objections ascribing it to e–e collisions, let us consider the relevance of the expression previously used7 for the T-square prefactor:

$$A=\frac{\hslash }{{{\rm{e}}}^{2}}{\left(\frac{{k}_{{\rm{B}}}}{{E}_{{\rm{F}}}}\right)}^{2}{l}_{{\rm{quad}}}$$
(2)

Here, is Planck’s constant divided by 2π, e is the electron charge, kB is the Boltzmann constant and lquad is a material-dependent characteristic length, which arises uniquely due to a dimensional analysis. Such an examination requires to know EF at a given n. For this, we performed a detailed study of quantum oscillations.

### Resistive quantum oscillations

Figure 3a, b shows that in presence of magnetic field, resistivity shows quantum oscillations with a single frequency for both in-plane and out-of-plane orientations of the magnetic field. When n ≈ 4.3 × 1018 cm−3, the oscillation frequency is FHc ≈  51 T and FHab ≈  93 T. This implies that the Fermi surface is an ellipsoid at the center of the Brillouin zone. The Fermi surface anisotropy $$\alpha =\frac{{F}_{{\rm{H}}\parallel {\rm{ab}}}}{{F}_{{\rm{H}}\parallel {\rm{c}}}}=1.8$$. This is in excellent agreement with our theoretical calculations (Fig. 1b), which finds that the dispersion along Γ-X and Γ-Z with the value of anisotropy (α) differ by 1.75. Similar data for other samples are presented in the Supplementary Fig. 4 and summarized in Supplementary Table 3.

The volume of the ellipsoidal Fermi pocket implies a carrier concentration nSdH = 3.8 × 1018 cm−3, close to the Hall carrier concentration. The numbers remain close to each other when n > 1018 cm−3 (see Supplementary Table 3). At very low densities, nSdH starts to fall below n, which may indicate that the single Fermi sea begins to fall apart to isolated lakes due to insufficient homogeneity in doping.

The small size of the Fermi surface excludes the possible occurrence of Umklapp event, which requires a Fermi wave vector larger than one-fourth of the smallest reciprocal lattice vector, G. For Bi2O2Se, the smallest G is along c-axis, $${{\bf{G}}}_{{\rm{c}}}=\frac{2\pi }{c}\approx 5.17$$ nm−1, given the lattice constant c = 12.16 Å of the tetragonal unit cell. Then the threshold carrier density for Umklapp scattering (nU) can be estimated to be $${n}_{{\rm{U}}}=\frac{1}{3{\pi }^{2}}{k}_{{\rm{Fa}}}^{2}{k}_{{\rm{Fc}}}=3\times 1{0}^{19}$$ cm−3 with $${k}_{{\rm{Fc}}}=\frac{{{\bf{G}}}_{{\rm{c}}}}{4}$$ and $$\alpha =\frac{{k}_{{\rm{Fc}}}}{{k}_{{\rm{Fa}}}}\approx 1.8$$ (kFc and kFa are Fermi wave vector along c-axis and a-axis). For n < nU, no Umklapp scattering is expected, which includes our range of study and beyond.

Figure 3c shows how quantum oscillations are damped with increasing temperature. This allows us to extract the effective mass m* using the Lifshitz–Kosevich (L–K) formula:

$${R}_{{\rm{L}}}=\frac{X}{\sinh (X)},\quad X=\frac{\eta T{m}^{* }}{H}$$
(3)

where $$\eta =\frac{2{\pi }^{2}{k}_{{\rm{B}}}}{{\rm{e}}\hslash }$$. We find an in-plane effective electron mass $${m}_{{\rm{H}}\parallel {\rm{c}}}^{* }\approx 0.17{m}_{{\rm{e}}}$$, in agreement with previous reports21,22,26, and an out-of-plane mass $${m}_{{\rm{H}}\parallel {\rm{ab}}}^{* }\approx 0.25{m}_{{\rm{e}}}$$ (me is the free electron mass). As expected, the ratio of these cyclotron masses is close to the ratio of the two ellipsoid axes.

Figure 3d shows $${E}_{{\rm{F}}}={\scriptstyle{\frac{{\left(\hslash {k}_{{\rm{Fab}}}\right)}^{2}}{2{m}_{{\rm{H}}\parallel {\rm{c}}}^{* }}}}$$ as a function of kFab (the in-plane wave vector).  Each    data point represents a different sample. As seen in Fig. 3e, the magnitude of the effective mass and Fermi surface anisotropy remain unchanged with doping. The data imply that the conducting band dispersion is parabolic. We also note that the effective mass resolved here (m* = 0.17 ± 0.1me, see Table S1) is only 1.3 ± 0.1 larger than the DFT calculated bare mass of mb = 0.125me. In contrast, in SrTiO3, band dispersion of the lower band is nonparabolic and there is a significant mass enhancement due to coupling to phonons7,11. This comparison highlights the simplicity of the case of Bi2O2Se where any polaronic effect seems to be absent.

### Scaling between A and EF

With the help of Fig. 3d, we can map n to EF and translate the data of Fig. 2g in a new figure (Fig. 4a), which compares the evolution of A with EF in Bi2O2Se and in SrTiO3−δ. Remarkably, the two sets of data join each other. Because of the lightness of its electrons, Bi2O2Se has a Fermi energy ten times higher than SrTiO3−δ at the same carrier density.

In Fig. 4b, we include the new Bi2O2Se data in a universal plot of A v.s. EF. The data for other materials are taken from references7,8,30 and are summarized in Supplementary Table 46. Note that for all anisotropic conductors including Bi2O2Se, the plot shows the prefactor in the plane with the higher conductivity.

This plot is an extension of the original Kadowaki–Woods plot4 to dilute Fermi liquids. In a dense metal, the electronic-specific heat (in molar units) is an accurate measure of the Fermi energy. In a dilute metal, on the other hand, the molar units for atoms and electrons differ by orders of magnitude and therefore, one cannot use the electronic-specific heat as a measure of the Fermi energy.

Kadowaki and Woods observed originally that correlation between A and γ2 in heavy-electron metals, such A/γ2 ≈ 10 μΩ.cm.K2.mol2.J−24 and contrasted it with a similar ratio noticed by Rice in elemental metals (A/γ2 ≈ 0.4 μΩ.cm.K2.mol2.J−2)3, where γ is the electronic-specific heat coefficient. Subsequent studies31 brought new data indicating that these ratios define two rough and lower boundaries (and recently even 3He has been shown to be on the Kadowaki–Woods plot32).

According to Fig. 4b, this 25-fold difference in the A/γ2 magnitude is equivalent to a statement on the boundaries of lquad, which lies between 1.6 and 40 nm across systems whose carrier concentration and Fermi energy differ by many orders of magnitude. This is in agreement with a recent observation by Kurita et al.30. They put under scrutiny the correlation between A and the low-temperature slope of the Seebeck efficient (which remains a measure of the Fermi energy even in dilute systems33) and found that in a variety of systems lquad ≈ 4 nm30.

lquad is a phenomenological quantity coming out of dimensional analysis. Nevertheless, it is well-defined and equal to the product of the Fermi wave vector and the cross-section of electron–electron collision7,34. This implies that its boundaries are meaningful and beg for an explanation.

## Discussion

Most previous theoretical attempts focus on isolated cases and did not seek a global scenario. Let us consider briefly their relevance to our data. One scenario for T2 resistivity, proposed decades ago, invokes inelastic electron-impurity scattering and its interplay with electron–phonon interaction35,36,37. Such an effect has been reported in several impure metals. It appears too weak to account for the T2 term in Bi2O2Se where $$\frac{A}{{\rho }_{0}} \sim 1{0}^{-3}$$ K−2. Pal and co-authors proposed that non-Galilean invariant FLs can display T2 resistivity even in the absence of Umklapp events38. However, this scenario for T2 resistivity does not expect it in a systems with parabolic dispersion such as Bi2O2Se. Quantum interference near a ferromagnetic quantum critical point (QCP) can induce a resistivity proportional to $${T}^{2}{\rm{ln}}T$$39. This is inapplicable to Bi2O2Se, which is not close to any QCP. Lucas pointed to hydrodynamic flow of electrons in random magnetic fields as a possible source of T2 resistivity13 in SrTiO3, speculating that oxygen vacancies can be magnetic there. Its relevance to a non-magnetic systems such as Bi2O2Se is quite unlikely.

In summary, we find that the fermiology of dilute metallic Bi2O2Se is such that interband or Umklapp scattering cannot happen. Nevertheless, there is a T2 resistivity unambiguously caused by electron–electron scattering. We find a universal link between Fermi energy and the prefactor of T-square resistivity, which persists across various Fermi liquids. We conclude that a proper understanding of the microscopic origin of T-square resistivity in Fermi liquids is missing.

## Methods

### Sample growth

Bi2O2Se poly-crystals were synthesized through solid state reaction with stoichiometric Bi (5N), Se (5N), and Bi2O3 (5N) powders of high purity. The mixed materials are sealed in an evacuated quartz tube and heated in an oven at 823K for 24 h. The single-crystalline phase was obtained through the chemical vapor transport (CVT) method by using poly-crystals as precursors. The sealed quartz tubes were placed in a horizontal furnace at a temperature gradient from 1123 to 1023 K over one week. The resulting single-crystals are shiny and air-stable. Note that, we didn’t use transport agents such as I2, in order to avoid unintentional doping. Samples with n below and above 1018 cm−3 are cleaved from two individual batches, respectively. Hence, we may expect a moderate inhomogeneous distribution of carrier concentrations in each sample.

### Experiments

X-ray diffraction patterns were performed using a Bruker D8 Advanced X-ray diffractometer with Cu Kα radiation at room temperature. The composition of samples was determined by an energy-dispersive X-ray (EDX) spectrometer affiliated to a Zeiss field emission scanning electron microscope (SEM). The transport measurements were done by a standard four-terminal method in Quantum Design PPMS-Dynacool equipped with 14T magnet. Ohmic contacts were obtained by evaporating gold pad to samples before attaching wires with silver paste.

### DFT calculations

All first-principles calculations were carried out using density functional theory (DFT) as implemented in Quantum Espresso40. The generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof revised for solids (PBEsol) type41 was used to describe the exchange-correlation energy. We used ultrasoft pseudopotentials from the Garrity, Bennett, Rabe, Vanderbilt (GBRV) high-throughput pseudopotential set42 and a plane-wave energy cutoff of 50 Ry and charge density cutoff of 250 Ry. The full phonon spectrum was calculated using the supercell approach as implemented in phonopy43 with the nonanalytical term correction at the Γ point included. As Bi is known for its strong spin-orbit coupling (SOC), the electronic band structure with SOC was calculated using the fully relativistic pseudopotentials taken from pslibrary (version 1.0.0)44. Both electronic band structure and phonon spectrum were calculated along the high-symmetry lines of the Brillouin zone of a body-centered tetragonal unit cell.